## The Annals of Probability

### Extremal theory for long range dependent infinitely divisible processes

#### Abstract

We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $\alpha$-Fréchet distribution and the skewed $\alpha$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $\alpha\in(0,\infty)$ and $\beta\in(0,1)$, with representations based on intersections of independent $\beta$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-\alpha$. The intriguing structure of these random sup-measures is due to intersections of independent $\beta$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $\beta$ increases to one. The results in this paper extend substantially previous investigations where only $\alpha\in(0,2)$ and $\beta\in(0,1/2)$ have been considered.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 2529-2562.

Dates
Revised: May 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205715

Digital Object Identifier
doi:10.1214/18-AOP1318

Mathematical Reviews number (MathSciNet)
MR3980927

Zentralblatt MATH identifier
07114723

#### Citation

Samorodnitsky, Gennady; Wang, Yizao. Extremal theory for long range dependent infinitely divisible processes. Ann. Probab. 47 (2019), no. 4, 2529--2562. doi:10.1214/18-AOP1318. https://projecteuclid.org/euclid.aop/1562205715

#### References

• Berman, S. M. (1964). Limit theorems for the maximum term in stationary sequences. Ann. Math. Stat. 35 502–516.
• Bertoin, J. (1999a). Subordinators: Examples and applications. In Lectures on Probability Theory and Statistics (Saint-Flour, 1997). Lecture Notes in Math. 1717 1–91. Springer, Berlin.
• Bertoin, J. (1999b). Intersection of independent regenerative sets. Probab. Theory Related Fields 114 97–121.
• Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York.
• Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge, MA.
• de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12 1194–1204.
• de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York.
• Doney, R. A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 451–465.
• Dwass, M. (1964). Extremal processes. Ann. Math. Stat. 35 1718–1725.
• Fisher, R. A. and Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample. In Mathematical Proceedings of the Cambridge Philosophical Society 24(2) 180–190. Cambridge Univ. Press, Cambridge, MA.
• Fitzsimmons, P. J., Fristedt, B. and Maisonneuve, B. (1985). Intersections and limits of regenerative sets. Z. Wahrsch. Verw. Gebiete 70 157–173.
• Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London.
• Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. (2) 44 423–453.
• Hawkes, J. (1977). Intersections of Markov random sets. Z. Wahrsch. Verw. Gebiete 37 243–251.
• Jung, P., Owada, T. and Samorodnitsky, G. (2017). Functional central limit theorem for a class of negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 45 2087–2130.
• Kabluchko, Z. (2009). Spectral representations of sum- and max-stable processes. Extremes 12 401–424.
• Kabluchko, Z. and Stoev, S. (2016). Stochastic integral representations and classification of sum- and max-infinitely divisible processes. Bernoulli 22 107–142.
• Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer, Berlin.
• Lacaux, C. and Samorodnitsky, G. (2016). Time-changed extremal process as a random sup measure. Bernoulli 22 1979–2000.
• Lamperti, J. (1964). On extreme order statistics. Ann. Math. Stat. 35 1726–1737.
• Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. Springer, New York.
• Letac, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications (Brunswick, Maine, 1984). Contemp. Math. 50 263–273. Amer. Math. Soc., Providence, RI.
• Mittal, Y. and Ylvisaker, D. (1975). Limit distributions for the maxima of stationary Gaussian processes. Stochastic Process. Appl. 3 1–18.
• Molchanov, I. (2005). Theory of Random Sets. Probability and Its Applications (New York). Springer, London.
• Molchanov, I. and Strokorb, K. (2016). Max-stable random sup-measures with comonotonic tail dependence. Stochastic Process. Appl. 126 2835–2859.
• O’Brien, G. L., Torfs, P. J. J. F. and Vervaat, W. (1990). Stationary self-similar extremal processes. Probab. Theory Related Fields 87 97–119.
• Owada, T. (2016). Limit theory for the sample autocovariance for heavy-tailed stationary infinitely divisible processes generated by conservative flows. J. Theoret. Probab. 29 63–95.
• Owada, T. and Samorodnitsky, G. (2015a). Maxima of long memory stationary symmetric $\alpha$-stable processes, and self-similar processes with stationary max-increments. Bernoulli 21 1575–1599.
• Owada, T. and Samorodnitsky, G. (2015b). Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43 240–285.
• Prudnikov, A. P., Brychkov, Yu. A. and Marichev, O. I. (1990). Integrals and Series. Vol. 3. More special functions. Gordon and Breach, New York. Translated from the Russian by G. G. Gould.
• Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.
• Resnick, S., Samorodnitsky, G. and Xue, F. (2000). Growth rates of sample covariances of stationary symmetric $\alpha$-stable processes associated with null recurrent Markov chains. Stochastic Process. Appl. 85 321–339.
• Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23 1163–1187.
• Rosiński, J. and Samorodnitsky, G. (1993). Distributions of subadditive functionals of sample paths of infinitely divisible processes. Ann. Probab. 21 996–1014.
• Rosiński, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoulli 2 365–377.
• Sabourin, A. and Segers, J. (2017). Marginal standardization of upper semicontinuous processes. With application to max-stable processes. J. Appl. Probab. 54 773–796.
• Salinetti, G. and Wets, R. J.-B. (1981). On the convergence of closed-valued measurable multifunctions. Trans. Amer. Math. Soc. 266 275–289.
• Salinetti, G. and Wets, R. J.-B. (1986). On the convergence in distribution of measurable multifunctions (random sets), normal integrands, stochastic processes and stochastic infima. Math. Oper. Res. 11 385–419.
• Samorodnitsky, G. (2004). Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes. Ann. Probab. 32 1438–1468.
• Samorodnitsky, G. (2005). Null flows, positive flows and the structure of stationary symmetric stable processes. Ann. Probab. 33 1782–1803.
• Samorodnitsky, G. (2016). Stochastic Processes and Long Range Dependence. Springer Series in Operations Research and Financial Engineering. Springer, Cham.
• Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge, MA. Translated from the 1990 Japanese original. Revised by the author.
• Simon, T. (2014). Comparing Fréchet and positive stable laws. Electron. J. Probab. 19 Article ID 16.
• Stoev, S. A. and Taqqu, M. S. (2005). Extremal stochastic integrals: A parallel between max-stable processes and $\alpha$-stable processes. Extremes 8 237–266.
• Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750–783.
• Vervaat, W. (1997). Random upper semicontinuous functions and extremal processes. In Probability and Lattices. CWI Tract 110 1–56. Centre for Mathematics and Computer Science, Amsterdam.